Some Estimates of Schrödinger-Type Operators with Certain Nonnegative Potentials

where the potential V belongs to Bq1 for q1 ≥ n/2. We are interested in the L boundedness of the operator∇4H−1, where the potential V satisfies weaker condition than that in 5, Theorem 1, 2 . The estimates of some other operators related to Schrödinger-type operators can be found in 2, 5 . Note that a nonnegative locally L integrable function V on R is said to belong to Bq 1 < q < ∞ if there exists C > 0 such that the reverse Hölder inequality ( 1 |B| ∫


Introduction
In recent years, there has been considerable activity in the study of Schr ödinger operators see 1-4 .In this paper, we consider the Schr ödinger-type operator where the potential V belongs to B q 1 for q 1 ≥ n/2.We are interested in the L p boundedness of the operator ∇ 4 H −1 , where the potential V satisfies weaker condition than that in 5, Theorem 1, 2 .The estimates of some other operators related to Schr ödinger-type operators can be found in 2, 5 .Note that a nonnegative locally L q integrable function V on R n is said to belong to B q 1 < q < ∞ if there exists C > 0 such that the reverse H ölder inequality holds for every ball B in R n .

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It follows from 3 that the B q class has a property of "self-improvement", that is, if V ∈ B q , then V ∈ B q ε for some ε > 0.
We now give the main results for the operator ∇ 4 H −1 in this paper.
Theorem 1.1.Suppose V ∈ B q 1 , q 1 ≥ n/2.Then for 1 < p ≤ q 1 /2 there exists a positive constant C p such that By the proof of Theorem 1.1, we obtain the following weak-type estimate.
Theorem 1.2.Suppose V ∈ B q 1 , q 1 ≥ n/2.Then for 1 < p ≤ q 1 /2 there exists a positive constant C 1 such that Under a stronger condition on the potential V , Sugano 5 has obtained the following proposition.
Proposition 1.3.Suppose V ∈ B n/2 and there exists a constant C such that V x ≤ Cm x, V 2 .Then for 1 < p < ∞ there exists a positive constant C p such that 1.5 As a direct consequence of our L p estimates, we have the following corollary.
Throughout this paper, unless otherwise indicated, we will use C to denote constants, which are not necessarily the same at each occurrence.By A ∼ B, we mean that there exist constants C > 0 and c > 0 such that c ≤ A/B ≤ C.

The auxiliary function m x, V and estimates of fundamental solution
In this section, we firstly recall the definition of the auxiliary function m x, V and some lemmas about the auxiliary function m x, V which have been proven in 3 .
Lemma 2.1.If V ∈ B q , q > 1, then the measure V x dx satisfies the doubling condition, that is, there exists C > 0 such that Assume that V ∈ B q 1 , q 1 ≥ n/2.The auxiliary function m x, V is defined by V y dy 1.

2.4
Moreover, Lemma 2.4.There exists l 0 > 0 such that for any x and y in R n ,

2.6
In particular, m x, V ∼ m y, V , if |x − y| < C/m x, V .

2.8
Refer to 3 for the proof of the above lemmas.

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The next lemma has been obtained by Tao and Wang in 6 .
In order to prove Theorem 1.1, we need to give the estimates of the fundamental solution of H. Zhong has established the estimates of the fundamental solution of H in 2 when V x is a nonnegative polynomial.Recently, Sugano 5 has obtained the polynomial decay estimates of the fundamental solution of H under a weaker condition on V in the following theorem.
Theorem 2.8.Assume V ∈ B n/2 and let Γ H x, y be the fundamental solution of H.For any positive integer N, there exists a constant C N such that 2.10

Proof of the main results
In this section, we will prove Theorems 1.1 and 1.2.
Theorem 3.1.Suppose V ∈ B q 1 , q 1 ≥ n/2.Then for 1 < p ≤ q 1 /2 there exists a positive constant C p such that Γ H x, y f y dy.

3.2
We need to show that where r 1/m x, V .Because of the self-improvement of the B q 1 class, V ∈ B q 0 for some q 0 > q 1 , we have where 1/q 2/q 0 1.Thus, V x q 0 m x, V n−2q 0 dx dy.

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Hence, we have proved that for some q 0 > q 1 ≥ n/2, By choosing s 2, α 4, and r 1/m x, V in Lemma 2.7, we immediately have 3.9 Thus,

3.10
Therefore, by using interpolation we have

3.11
Then we deal with u 2 .For 1 < p ≤ q 0 /2, by the H ölder inequality, where r 1/m x, V and we apply the second inequality for s 0 and α 4 in Lemma 2.7 to the last step.

3.13
Fix y ∈ R n and let R 1/m y, V .By Lemmas 2.4, 2.6, and 2.7, From this, we have

3.15
Thus the theorem is proved.Now we give the proof of Theorem 1.1.
Proof of Theorem 1.1.Suppose V ∈ B q 1 for some q 1 ≥ n/2.By Theorem 3.1, we have

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Proof of Theorem 1.2.Note that ∇ 4 −Δ −2 satisfies

3.19
Thus, by the proof of Theorem 1.1, 3.20